Mobile sub-millimeter micro-robots have demonstrated untethered motion and transport of cargo in remote, confined or enclosed environments. However, limited by simple design and actuation, they lack remotely-actuated on-board mechanisms required to perform complex tasks such as object assembly. A flexible patterned magnetic material which allows internal actuation, resulting in a mobile micro-gripper which is driven and actuated by magnetic fields, is introduced here. By remotely controlling the magnetization direction of each micro-gripper arm, a gripping motion which can be combined with locomotion for precise transport, orientation, and programmable three-dimensional assembly of micro-parts in remote environments is demonstrated. This allows the creation of out-of-plane 3D structures and mechanisms made from several building blocks. Using multiple magnetic materials in each micro-gripper, the addressable actuation of gripper teams for parallel, distributed operation is also demonstrated. These mobile micro-grippers can potentially be applied to 3D assembly of heterogeneous meta-materials, construction of medical devices inside the human body, the study of biological systems in micro-fluidic channels, 3D micro-device prototyping or desktop micro-factories.